Manin obstruction

In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction is attached to a geometric object X which measures the failure of the Hasse principle for X: that is, if the value of the obstruction is non-trivial, then X may have points over all local fields but not over a global field.

For abelian varieties the Manin obstruction is just the Tate-Shafarevich group and fully accounts for the failure of the local-to-global principle. There are however examples, due to Skorobogatov, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points.

References

• Serge Lang (1997). Survey of Diophantine geometry. Springer-Verlag, 250-258. ISBN 3-540-61223-8. 
• Alexei N. Skorobogatov (1999). "Beyond the Manin obstruction". Invent. Math. 135 (2): 399-424.

• Alexei Skorobogatov (2001). Torsors and rational points, 1-7,112. ISBN 0521802377.